Forecasting with DSGE Models

Transcription

1 Forecasting with DSGE Models Kai Christoffel, Günter Coenen and Anders Warne DG-Research, European Central Bank December, 9 Abstract: In this paper we review the methodology of forecasting with log-linearised DSGE models using Bayesian methods. We focus on the estimation of their predictive distributions, with special attention being paid to the mean and the covariance matrix of h-steps ahead forecasts. In the empirical analysis, we examine the forecasting performance of the New Area-Wide Model (NAWM) that has been designed for use in the macroeconomic projections at the European Central Bank. The forecast sample covers the period following the introduction of the euro and the out-of-sample performance of the NAWM is compared to nonstructural benchmarks, such as Bayesian vector autoregressions (BVARs). Overall, the empirical evidence indicates that the NAWM compares quite well with the reduced-form models and the results are therefore in line with previous studies. Yet there is scope for improving the NAWM s forecasting performance. For example, the model is not able to explain the moderation in wage growth over the forecast evaluation period and, therefore, it tends to overestimate nominal wages. As a consequence, both the multivariate point and density forecasts using the log determinant and the log predictive score, respectively, suggest that a large BVAR can outperform the NAWM. Keywords: Bayesian inference, DSGE models, euro area, forecasting, open-economy macroeconomics, vector autoregression. JEL Classification Numbers: C, C3, E3, E37.. Introduction Since the turn of the century, we have witnessed the development of a new generation of dynamic stochastic general equilibrium (DSGE) models that build on explicit micro-foundations with optimising agents. Major advances in estimation methodology allowed estimating variants of these models that are able to compete, in terms of data coherence, with more standard time series models, such as vector autoregressions (VARs); see, among others, Christiano, Eichenbaum, and Evans (5), Smets and Wouters (3, 7), Adolfson, Laséen, Lindé, and Villani (7), and Christoffel, Coenen, and Warne (). Accordingly, the new generation of DSGE models provides a framework that appears particularly suited for evaluating the consequences of alternative macroeconomic policies. Efforts have also been undertaken to bring these models to the forecasting arena. Results in Smets and Wouters () suggest that the new generation of closed-economy DSGE models compare well with conventional forecasting tools such as VAR models; see also Edge, Kiley, Note: The paper is in preparation for appearing as a chapter in an Oxford Handbook on Economic Forecasting, edited by Michael P. Clements and David F. Hendry. We are very grateful to Marta Bańbura who has estimated and computed the forecasts for the two large Bayesian VAR models we have used in the paper. We have received valuable comments from participants at the Nottingham workshop on DSGE modelling, December 9, and seminar participants at the Humboldt University in Berlin. We are particularly grateful for discussions with and comments from Richard Anderson, Michael Burda and Alexander Meyer-Gohde. The views expressed in this paper do not necessarily reflect those of the European Central Bank.

2 and Laforte (9). Similarly, the study by Adolfson, Lindé, and Villani (7) shows that also open-economy DSGE models can compete well with reduced-form models. However, it is worth recalling that the study by Del Negro, Schorfheide, Smets, and Wouters (7) finds evidence that the Smets and Wouters model is misspecified when estimated on postwar U.S. data and when applying the goodness-of-fit tools proposed in that study. Moreover, they show that this DSGE model is outperformed by a so-called DSGE-VAR in terms of out-of-sample point forecast accuracy. Against this background, the goal of the current paper is to review and illustrate the methodology of forecasting with DSGE models using Bayesian methods. We limit the scope of the paper to log-linearised DSGE models; and, hence, we neither consider DSGE-VARs, as in Del Negro and Schorfheide (), nor do we consider DSGE models based on higher-order approximations, as in Fernández-Villaverde and Rubio-Ramírez (5). As regards the initial steps of forecasting with DSGE models, Sargent (99) was amongst the first to point out that a log-linearised DSGE model can be cast in the familiar state-space form, where the observed variables are linked to the model variables (and possibly to measurement errors) through the measurement equation. At the same time, the state equation provides the reduced form of the DSGE model, mapping current model variables to their lags and the underlying i.i.d. shocks, where the reduced form is obtained by solving for the expectation terms in the structural form of the model using a suitable method; see, e.g., Blanchard and Kahn (9), Anderson and Moore (95), Klein (), or Sims (). The Kalman filter can thereafter be used to compute the value of the log-likelihood function for any value of the model parameters when a (unique) solution of the DSGE model exists. A classical approach to the estimation of these parameters would then be to maximise the log-likelihood function with numerical methods. A Bayesian approach would instead complement the likelihood with a prior distribution for the parameters and estimate the posterior mode through numerical optimisation, or other properties of the posterior distribution via Markov Chain Monte Carlo (MCMC) methods. In this paper, we shall discuss an algorithm for estimating the predictive distribution of the observed variables based on draws from the posterior distribution of the DSGE model parameters and simulation of future paths for the variables with the model. The general method, called sampling the future, was first suggested for univariate time series models by Thompson and Miller (9). Their variant was simplified and adapted to VAR models by Villani (). The particular version of the algorithm that can be used for state-space models was suggested in Adolfson, Lindé, and Villani (7). In case the forecast evaluation exercise only requires moments from the predictive distribution, such as the mean and the covariance, then the simulation algorithm is not necessary. Estimation of such moments can instead be achieved by properly combining population moments for fixed parameter values with draws from the posterior distribution and, thus, without sampling the future via the model. However, if we also wish to estimate, e.g., quantiles, confidence intervals or the probability that the variables reach some

3 barrier, then the simulation algorithm may prove useful. We note that the algorithm does not rely on a particular posterior sampler. It only requires that the draws characterise the posterior distribution of the parameters. We illustrate these tools by applying them to a particular DSGE model. We have selected the New Area-Wide Model (NAWM), developed at the European Central Bank (ECB), which is designed for use in the (Broad) Macroeconomic Projection Exercises regularly undertaken by ECB/Eurosystem staff and for policy analysis. The specification of the NAWM was influenced by both economic and statistical criteria. For example, impulse-response functions and forecast-error-variance decompositions were used for assessing alternative specifications from an economic perspective, while the marginal likelihood and comparisons between model-based sample moments and estimates from the data only were applied as statistical model evaluation criteria. In addition, a small forecast evaluation exercise was conducted, but it was treated as one among many criteria for assessing the performance of the model. Here we extend the forecast evaluation exercise to the full set of the NAWM s endogenous variables. The forecast sample covers the period following the introduction of the euro and we shall study both point and density forecasts from up to quarters ahead. The DSGE model forecasts are compared to those from a VAR and three Bayesian VARs (BVARs), as well as the naïve random walk and (sample) mean benchmarks. We shall also consider different subsets of the observed variables included in the NAWM, as well as different transformations of these variables. The remainder of the paper is organised as follows. Section sketches the NAWM, while Section 3 reports on our implementation of Bayesian inference methods and on some selected estimation results for the NAWM. Section first discusses how the predictive distribution of a DSGE model can be estimated, and it then presents the alternative forecasting models that are used in the empirical analysis. Section 5 covers the forecast evaluation of the NAWM, focusing first on point forecasts and then on density forecasts. Section summarises the main findings of the paper and concludes.. The New Area-Wide Model of the Euro Area In this section we provide a brief overview of the NAWM to set the stage for our review of the methodology for forecasting with log-linearised DSGE models. The NAWM is a micro-founded open-economy model of the euro area designed for use in the ECB/Eurosystem staff projections and for policy analysis; see Christoffel, Coenen, and Warne () for a detailed description of the NAWM s structure. Its development has been guided by a principal consideration, namely to provide a comprehensive set of core projection variables, including a number of foreign variables, which, in the form of exogenous assumptions, play an important role in the projections. As a consequence, the scale of the NAWM compared with a typical DSGE model is rather large, and it is estimated on macroeconomic time series. 3

4 .. A Bird s Eye View on the Model The NAWM features four classes of economic agents: households, firms, a fiscal authority and a monetary authority. Households make optimal choices regarding their purchases of consumption and investment goods, they supply differentiated labour services in monopolistically competitive markets, they set wages as a mark-up over the marginal rate of substitution between consumption and leisure, and they trade in domestic and foreign bonds. As regards firms, the NAWM distinguishes between domestic producers of tradable differentiated intermediate goods and domestic producers of three types of non-tradable final goods: a private consumption good, a private investment good, and a public consumption good. The intermediate-good firms use labour and capital as inputs to produce their differentiated goods, which are sold in monopolistically competitive markets domestically and abroad. Accordingly, they set different prices for domestic and foreign markets as a mark-up over their marginal costs. The final-good firms combine domestic and foreign intermediate goods in different proportions, acting as price takers in fully competitive markets. The foreign intermediate goods are imported from producers abroad, who set their prices in euro, allowing for an incomplete exchange-rate pass-through. A foreign retail firm in turn combines the exported domestic intermediate goods, where aggregate export demand depending on total foreign demand. Both households and firms face nominal and real frictions, which have been identified as important in generating empirically plausible dynamics. Real frictions are introduced via external habit formation in consumption and through generalised adjustment costs in investment, imports and exports. Nominal frictions arise from staggered price and wage-setting à la Calvo (93), along with (partial) dynamic indexation of price and wage contracts. In addition, there exist financial frictions in the form of domestic and external risk premia. The fiscal authority purchases the public consumption good, issues domestic bonds, and levies different types of distortionary taxes. Nevertheless, Ricardian equivalence holds because of the simplifying assumption that the fiscal authority s budget is balanced each period by means of lump-sum taxes. The monetary authority sets the short-term nominal interest rate according to a Taylor-type interest-rate rule, with the objective of stabilising inflation in line with the ECB s definition of price stability. The NAWM is closed by a rest-of-the-world block, which is represented by a structural vectorautoregressive (SVAR) model determining a small set of foreign variables: foreign demand, foreign prices, the foreign interest rate, foreign competitors export prices and the price of oil. The SVAR model does not feature spill-overs from the euro area, in line with the treatment of the foreign variables as exogenous assumptions in the projections... Some Key Model Equations To better understand the cross-equation restrictions implied by the NAWM s structure, it is instructive to look at some key behavioural equations in their log-linearised form. We focus on

5 those equations most closely related to the set of observed variables that form the basis of the forecasting performance evaluation in Section 5; namely, private consumption, investment, imports and exports, the private consumption and the import deflator, wages and employment, the short-term nominal interest rate and the real effective exchange rate. Real GDP and the GDP deflator are obtained from the model s aggregate resource constraint in real and in nominal terms, respectively. In order to derive the log-linearised equations, the NAWM is first cast into stationary form. To this end, all real variables are measured in per-capita terms and scaled by trend labour productivity z t. This variable is assumed to follow a random walk with stochastic drift and defines the model s balanced growth path. Similarly, we normalise all nominal variables with the price of the consumption good P C,t. For example, we use c t = C t /z t to denote the stationary level of per-capita consumption, while we use p I,t = P I,t /P C,t to represent the stationary relative price of the investment good. We then proceed with the log-linearisation of the transformed NAWM around its deterministic steady state, where the logarithmic deviation of a variable from its steady-state value is denoted by a hat ( ). For example, the log-deviation from steady state for the scaled consumption variable is ĉ t = log(c t /c). With these conventions, private consumption ĉ t is characterised by an intertemporal optimality condition (Euler equation), which relates the log-difference of current and expected future consumption to the ex-ante real interest rate, r t E t [ π C,t+ ], noting that the specific form of the households utility function, with additive habits and habit formation parameter κ, implies that also lagged consumption enters the consumption equation: ĉ t = Here, ǫ RP t + κgz E t [ĉ t+ ] + κg z + κg z ĉ t κg ( z + κgz ) r t E t [ π C,t+ ] + ǫ RP t ( + κgz E t [ĝ z,t+ ] κgz ĝ z,t ). denotes a risk-premium shock, which drives an exogenous wedge between the riskless interest rate set by the monetary authority and the effective interest rate faced by households. The expected quasi-difference of trend labour productivity growth, E t [ĝ z,t+ ] κg z ĝ z,t, enters as an additional term because of the scaling of the consumption variable with the level of trend productivity, where g z denotes the steady-state value of g z,t = z t /z t. Investment î t is characterised by an equation with a similar structure. The intertemporal price of investment is given by the log-difference of Tobin s Q the discounted sum of expected future returns of the existing capital stock, with discount factor β and the price of newly installed capital goods, Q t p I,t : î t = β ] + β E t [ît+ + + β ît + γ I gz ( + β) ( Qt p I,t + ǫ I t) + + β (β E t [ ĝ z,t+ ] ĝ z,t ). () () 5

6 The intertemporal price of investment is shifted by an investment-specific technology shock ǫ I t, which affects the efficiency of newly installed capital goods. The lagged investment term reflects the existence of adjustment costs related to incremental changes in investment, with sensitivity parameter γ I. Private consumption and investment are composed of bundles of domestic and imported intermediate goods, îm C t and îm I t. The demand for these import bundles depends on the total demand for the consumption good, q C t = ĉ t, and the investment good, q I t = î t, respectively. Suppressing the consumption and investment superscripts for the sake of simplicity and focusing on the generic form of the import demand equation, the share of imports in total demand is then obtained as a function of the price of the imported intermediate-goods bundle relative to the price of the generic final good, p IM,t p t : ( îm t = µ p IM,t p t Γ ) IM,t + q t. (3) Here, the parameter µ represents the price elasticity of import demand. As in the case of investment, adjustment costs are incurred which, in their generic form Γ IM,t, dampen the influence of changes in the relative price of imports on import demand. The demand for euro area exports x t is determined in a similar way as a share of euro area foreign demand ŷ t. This share varies with the price of euro area exports (translated into foreign currency with the real effective exchange rate ŝ t, denominated in terms of the GDP deflator p Y,t ) relative to the price of exports of the euro area s competitors, p X,t ŝ t p Y,t p c X,t : ( p x t = µ X,t ŝ t p Y,t p c X,t Γ ) X,t + ŷt + ν t, () where the parameter µ denotes the price elasticity of exports. The term Γ X,t represents generic adjustment costs, and the term ν t is an exogenous shock to foreign export preferences. Consumer prices are determined as a combination of the aggregate prices of the domestically produced and the imported intermediate goods, p H,t and p IM,t. The evolution of these prices is governed, in generic form, by forward-looking Phillips-curve equations according to which the rate of price inflation π t gradually adjusts in response to fluctuations in real marginal costs mc t, subject to an exogenous price mark-up shock ϕ t : π t = β + βχ E t [ π t+ ] + χ + βχ π ( βξ)( ξ) t + ξ ( + βχ) ( mc t + ϕ t ). (5) This equation derives from the typical Calvo assumption that firms can only infrequently re-set their prices optimally, namely with probability ξ. Those firms which are not permitted to do so are allowed to index their prices to past inflation π t with indexation parameter χ. Real wages and hours worked are the key labour-market variables in the NAWM. Real wages ŵ t adjust gradually according to a forward-looking Phillips-curve equation which closes the gap between the after-tax real wage ŵ τ t and the marginal rate of substitution mrs t, subject to an

7 exogenous wage mark-up shock ϕ W t : ŵ t = β + β E t [ŵ t+ ] + + β ŵt + β + β E t [ π C,t+ ] () + βχ W + β π C,t + χ W + β π C,t ( βξ W )( ξ W ) ( + β)ξ W ( + ϕw ϕ W ζ) (ŵτ t mrs t ϕ W t ). As in the case of the price Phillips curves, the parameters ξ W and χ W denote, respectively, the Calvo adjustment probability for (nominal) wages and the degree of indexation to past consumer price inflation π C,t. The parameter ϕ W denotes the steady-state wage markup and ζ the Frisch elasticity of labour supply. Since there exist no reliable data for hours worked in the euro area, we rely on employment data and relate the employment variable Êt to the NAWM s unobserved hours-worked variable N t by an auxiliary equation following Smets and Wouters (3), Ê t = β + β E t[êt+] + + β Êt + ( βξ E )( ξ E ) ( ) Nt ( + β)ξ Êt. (7) E Here, the parameter ξ E determines the sensitivity of employment with respect to hours worked, similar to the role of the Calvo parameters in the price and wage Phillips curves. The monetary authority sets the short-term nominal interest rate r t according to a simple Taylor-type interest-rate rule, where the parameter φ R represents the degree of interest-rate smoothing and the parameters φ Π, φ Π and φ Y determine the sensitivity of the interest-rate response to, respectively, consumer price inflation, the change in inflation and real GDP growth (relative to trend productivity growth): r t = φ R r t + ( φ R )φ Π π C,t + φ Π ( π C,t π C,t ) + φ Y (ŷ t ŷ t ) + η R t. () The term η R t denotes a serially uncorrelated monetary policy shock. Finally, the real effective exchange rate ŝ t is determined by a risk-adjusted uncovered interest parity condition: ŝ t = E t [ ŝ t+ ] r t + r t ǫrp t [ + E t πy,t+ π Y,t+ ] γb ŝ B,t+ ǫ RP t, (9) where r t and π Y,t+ denote the foreign interest rate and foreign inflation, respectively. The last two terms represent an external risk premium. It is composed of an endogenous component related to the net holdings of foreign bonds, ŝ B,t+ with sensitivity γ B, and an exogenous shock ǫ RP t. The NAWM s log-linearised equations, including the equations presented above, can be easily cast in state-space form, where the state equation corresponds to the reduced-form solution of the model, which we obtain using the AIM algorithm developed in Anderson and Moore (95). The observed variables are related to the model s state variables through an appropriate measurement equation. 7

8 3. Bayesian Estimation of DSGE Models We adopt the empirical approach outlined in Smets and Wouters (3) and An and Schorfheide (7) and estimate the NAWM employing Bayesian inference methods. This involves obtaining the posterior distribution of the model s parameters based on its log-linear state-space representation using the Kalman filter. For the empirical analyses, we use YADA, a Matlab programme for Bayesian estimation and evaluation of DSGE models; see Warne (9). In the following we sketch the adopted approach and describe the data and the shock processes that we consider in its implementation. We then briefly report on the calibration of the model s steady state and present some selected estimation results. 3.. Methodology Employing Bayesian inference methods allows formalising the use of prior information obtained from earlier studies at both the micro and macro level in estimating the parameters of a possibly complex DSGE model. This seems particularly appealing in situations where the sample period of the data is relatively short, as is the case for the euro area. From a practical perspective, Bayesian inference may also help to alleviate the inherent numerical difficulties associated with solving the highly non-linear estimation problem. Formally, let p(θ m m) denote the prior distribution of the vector θ m Θ m with structural parameters for some model m M, and let p(y T θ m,m) denote the likelihood function for the observed data, Y T = {y,...,y T }, conditional on parameter vector θ m and model m. The joint posterior distribution of θ m for model m is then obtained by combining the likelihood function for Y T and the prior distribution of θ m, p(θ m Y T,m) p(y T θ m,m)p(θ m m), where denotes proportionality. The posterior distribution is typically characterised by measures of location, such as the mode or the mean, measures of dispersion, such as the standard deviation, or selected quantiles. Following Schorfheide (), we adopt an MCMC sampling algorithm to determine the joint posterior distribution of the parameter vector θ m. More specifically, we rely on the random-walk Metropolis algorithm with a Gaussian proposal density to obtain a large number of random draws from the posterior distribution of θ m. The posterior mode and the inverse Hessian matrix are computed by a standard numerical optimisation routine, namely Christopher Sims optimiser csminwel. As discussed in Geweke (999), Bayesian inference also provides a framework for comparing alternative and potentially misspecified models on the basis of their marginal likelihood. For a given model m the latter is obtained by integrating out the parameter vector θ m, p(y T m) = p(y T θ m,m)p(θ m m)dθ m. θ m Θ m

9 Thus, the marginal likelihood gives an indication of the overall likelihood of the observed data conditional on a model. To estimate the marginal likelihood one may use the modified harmonic mean estimator, suggested by Geweke (999); see also Geweke (5). An alternative estimator, suggested by Chib and Jeliazkov (), relies on rewriting Bayes theorem into the so-called marginal likelihood identity. The former estimator requires only draws from the posterior of θ m, while the latter also requires draws of these parameters from the proposal density. 3.. Data and Shock Processes In estimating the NAWM, we use times series for macroeconomic variables which feature prominently in the ECB/Eurosystem staff projections: real GDP, private consumption, total investment, government consumption, extra-euro area exports and imports, the GDP deflator, the consumption deflator, the extra-euro area import deflator, total employment, nominal wages per head, the short-term nominal interest rate, the nominal effective exchange rate, foreign demand, foreign prices, the foreign interest rate, competitors export prices, and the price of oil. All time series are taken from an updated version of the AWM database Fagan, Henry, and Mestre (5), except for the time series of extra-euro area trade data the construction of which is detailed in Dieppe and Warmedinger (7). The sample period ranges from 95Q to Q (using the period 9Q to 9Q as training sample). The last five variables are modelled using a structural VAR model, the estimated parameters of which are kept fixed throughout the estimation of the NAWM. Similarly, government consumption is specified by means of a univariate autoregressive (AR) model with fixed estimated parameters. For details, see Christoffel, Coenen, and Warne (), Section 3.. Prior to estimation, we transform real GDP, private consumption, total investment, extraeuro area exports and imports, the associated deflators, nominal wages per head, as well as foreign demand and foreign prices into quarter-on-quarter growth rates, approximated by the first difference of their logarithm. Furthermore, a number of additional transformations are made to ensure that variable measurement is consistent with the properties of the NAWM s balanced-growth path and in line with the underlying assumption that all relative prices are stationary. First, the sample growth rates of extra-euro area exports and imports as well as foreign demand are matched with the sample growth rate of real GDP by removing the sample growth rate differentials, reflecting the fact that trade volumes and foreign demand tend to grow at a significantly higher rate than real GDP. Second, for the logarithm of government consumption we remove a linear trend consistent with the NAWM s steady-state growth rate of. percent per annum which is assumed to have two components: labour productivity growth g z of roughly. percent and labour force growth of approximately. percent. The former is broadly in line with the average labour productivity growth over the sample period. Third, we take the logarithm of employment and remove a linear trend consistent with a steady-state labour force growth rate of. percent, noting that, in the absence of a reliable measure of hours worked, we use data on employment in the estimation. Fourth, we construct a measure of the 9

10 real effective exchange rate from the nominal effective exchange rate, the domestic GDP deflator and foreign prices (defined as a weighted average of foreign GDP deflators) and then remove the mean. Finally, competitors export prices and oil prices (both expressed in the currency basket underlying the construction of the nominal effective exchange rate) are deflated with foreign prices before unrestricted linear trends are removed from the variables. Figure shows the time series of the transformed variables for the sample period 95Q to Q. To ensure that the -step ahead covariance matrix in the likelihood function for the observed variables is non-singular, the NAWM features distinct structural shocks, several of which have been discussed in Section. above, plus the shocks in the AR and SVAR models for government consumption and the foreign variables, respectively. All shocks are assumed to follow first-order autoregressive processes, except for the monetary policy shock and the shocks in the AR and SVAR models, which are assumed to be serially uncorrelated. We recall in this context that assuming an autoregressive process for trend labour productivity growth g z,t referred to as the NAWM s permanent technology shock implies that all real variables, with the exception of hours worked and employment, share a common stochastic trend, in line with the model s balanced-growth property. In addition, we account for measurement error in extra-euro area trade data (both volumes and prices) in view of the fact that they are prone to revisions. We also allow for small errors in the measurement of real GDP and the GDP deflator to alleviate discrepancies between the national accounts framework underlying the construction of official GDP data and the NAWM s aggregate resource constraint Empirical Results An extensive discussion of the empirical implementation of the NAWM is beyond the scope of this paper, and the reader is thus referred to Christoffel, Coenen, and Warne () for details. Here we report selectively on the calibration of the model s steady state and the posterior distribution of some key estimated parameters, which is deemed helpful for understanding the model s forecasting performance analysed in Section 5. Regarding the NAWM s steady state, all real variables are assumed to evolve along a balancedgrowth path with a trend growth rate of percent per annum, which roughly matches average real GDP growth in our estimation sample. Since the steady-state growth rate for the labour force can be seen as a proxy for population growth, all quantities within the NAWM can be interpreted in per-capita terms once it has been accounted for. Consistent with the balancedgrowth assumption, we then calibrate key steady-state ratios of the model by matching their empirical counterparts over the sample period. For example, the expenditure shares of private consumption, total investment and government consumption are set to, respectively, 57.5, and.5 percent of nominal GDP, while the export and import shares are set to percent, ensuring balanced trade in steady state. On the nominal side the monetary authority s longrun (net) inflation objective Π is set equal to.9 percent at an annualised rate, consistent

11 with the ECB s quantitative definition of price stability of inflation being below, but close to percent. This implies that, within the NAWM, nominal wages grow with a steady-state rate of 3. percent, corresponding to the sum of trend labour productivity growth of. percent and the inflation objective of.9 percent. As to the choice of prior distributions for the NAWM s estimated parameters we follow Smets and Wouters (3) since their closed-economy model of the euro area is essentially nested within the NAWM. Our choice of prior distributions for the parameters concerning the NAWM s openeconomy dimension is informed by the priors employed in Adolfson, Laséen, Lindé, and Villani (7). Comparing the plots of the prior and posterior distributions we find that the observed data provide additional information for most parameters. A number of estimation results are noteworthy. First, the estimates of the parameters shaping the dynamics of domestic demand in response to the model s structural shocks the degree of habit formation in consumption, κ, and the investment adjustment cost parameter, γ I are broadly in line with those reported by Smets and Wouters. Second, on the nominal side, we observe that the estimate of the Calvo parameter constraining the frequency of price-setting decisions of domestic firms selling in home markets, ξ H, is rather high. Yet our posterior mode estimate of about.9 is comparable with a point estimate of about.9 for the Calvo parameter in the model of Smets and Wouters. The estimate implies that the NAWM s domestic Phillips curve is rather flat or, in other words, that the sensitivity of domestic inflation with respect to movements in real marginal cost is low. Similarly, the posterior mode estimate of the indexation parameter χ H is., suggesting a relatively low degree of inflation persistence. Third, regarding the interest-rate rule, we observe that the estimated response coefficients φ R, φ Π, φ Π and φ Y are rather close to the estimates reported in Smets and Wouters, despite the fact that the NAWM s interest-rate rule does not feature a response to the so-called flex-price output gap, unlike the rule considered by Smets and Wouters. Finally, regarding the properties of the structural shocks, none of the estimated shock processes appears excessively persistent. Figure depicts the prior and posterior distributions of the structural parameters κ, γ I, ξ H and χ H, and the response coefficients of the interest-rate rule, φ R, φ Π, φ Π and φ Y, using the full sample, whereas Figure 3 shows the sequence of the posterior mode estimates when the sample is updated recursively over the period following the introduction of the euro. Overall, the recursively updated posterior mode estimates reveal a rather high degree of stability. Yet the gradual upward shift of the Calvo parameter ξ H suggests that domestic inflation has become less sensitive to movements in marginal costs over time. The gradual fall in the indexation parameter χ H implies a diminishing degree of inflation persistence, which may be interpreted as an indication that the anchoring of inflation expectations has been strengthened with the introduction of the euro area.

12 . Bayesian Forecasting by Sampling the Future.. Estimating the Predictive Distribution of a DSGE Model Let θ Θ be a vector of parameters for the log-linearised DSGE model; to simplify notation we have omitted the model m index in this section. Given that a unique convergent solution exists at a particular value for the parameter vector, we can express the relationship between the model variables, defined as deviations from the steady state, and the parameters as a VAR system. Specifically, let η t be a q-dimensional vector with i.i.d. standard normal structural shocks (η t N(,I q )), while ξ t is an r-dimensional vector of model variables for t =,,...,T. The solution (reduced form) of a log-linearised DSGE model can now be represented by: ξ t = Fξ t + Bη t, t =,...,T, () where F and B are uniquely determined by θ. The observed variables are denoted by y t, an n-dimensional vector, which is linked to the model variables ξ t through the equation y t = A x t + H ξ t + w t, t =,...,T. () The k-dimensional vector x t is here assumed to be deterministic, while w t is a vector of i.i.d. normal measurement errors with mean zero and covariance matrix R. The measurement errors and the shocks η t are assumed to be independent, while the matrices A, H, and R are uniquely determined by θ. The system in () and () is a state-space model with ξ t being partially unobserved state variables when, for example, r > n. Equation () gives the state or transition equation and () the measurement or observation equation. Provided the number of measurement errors and structural shocks is large enough, we can calculate the likelihood function for the observed data Y T = {y,...,y T } via the Kalman filter; see, e.g., Hamilton (99) for details. The filter can also be used to estimate all unobserved variables in the model at the given value for θ. The predictive density of y T+,...,y T+H can be expressed as p ( ) y T+,...,y T+H Y T = p ( y T+,...,y T+H Y T,θ ) p ( ) θ Y T dθ, () θ Θ where p(θ Y T ) is the posterior density of θ based on the data available at time T. Since the integral in () cannot be evaluated analytically we can apply a numerical algorithm adapted by Adolfson, Lindé, and Villani (7) to state-space models; see also Thompson and Miller (9). That is: () Draw θ from p(θ Y T ); () Draw the state variables at time T from ξ T N(ξ T T,P T T ), where ξ T T is the filter estimate of ξ T and P T T is the covariance matrix of ξ T given θ and Y T ; (3) Simulate a path for the state variables from () using the drawn value for ξ T as initial value and a sequence of structural shocks η T+,...,η T+H drawn from N(,I q );

13 () Draw a sequence of measurement errors w T+,...,w T+H from N(,R) and compute the path for the observed variables y T+,...,y T+H using the measurement equation (); (5) Repeat steps - M times for the same θ; () Repeat steps -5 M times. The algorithm thus gives M = M M paths from the predictive distribution in (). Point and interval forecasts as well as quantiles can now be computed in a straightforward manner. However, it may be noted that if the forecast evaluation exercise only requires moments from the predictive distribution, such as the mean and the covariance matrix, then the above algorithm is not needed. The population mean of y T+h given Y T and θ is E [ y T+h Y T,θ ] = A x T+h + H F h ξ T T, h =,...,H. (3) To estimate the mean of the predictive distribution of y T+h we may simply compute the sample average of the right hand side of (3) for θ (i) p(θ Y T ), i =,...,M. By choosing M large enough, the numerical standard error of this estimator of E[y T+h Y T ] is negligible. Similarly, the covariance matrix of y T+h conditional on Y T and θ is C [ y T+h Y T,θ ] = H F h ( P T T F h ) h H + H F j BB ( F j ) H + R. () The first term on the right hand side represents state-variable uncertainty given θ, the second term reflects uncertainty due to the structural shocks, and the third the uncertainty due to measurement errors. Following Adolfson, Lindé, and Villani (7), the prediction covariance matrix of y T+h is given by C [ ] [ [ y T+h Y T = ET C yt+h Y T,θ ]] [ [ + C T E yt+h Y T,θ ]], (5) where E T and C T denote the expectation and covariance with respect to the posterior of θ at time T. The second term on the right hand side of (5) measures the impact that parameter uncertainty has on the h-steps ahead forecasts based on the population mean, while the first term can be decomposed into uncertainties due to unobserved state variables, structural shocks and measurement errors, where the dependence on the parameters has now been dealt with. The first term in (5) can be estimated by the sample average of C[y T+h Y T,θ (i) ] in () for the M draws from p(θ Y T ), while the second term can be estimated by the sample covariance matrix of E[y T+h Y T,θ (i) ] in (3) using these M draws. Again, we can choose M large enough such that the numerical standard errors of the estimators are negligible... Alternative Forecasting Models Sims (9) convincingly argued that vector autoregressions (VARs) provide a less restrictive environment for modelling macroeconomic time series than the large-scale structural macroeconometric models, based on incredible identifying assumptions, that were prevalent at the time. However, while VARs often provide a reasonably good fit of macroeconomic time series data, a 3 j=

14 problem with using them is that they are not parsimonious and, hence, the number of variables that can be included is limited by a lack of long time series. To overcome this problem in forecasting situations, the so called Minnesota prior (Doan, Litterman, and Sims, 9) makes use of the old idea of shrinkage, a flexible method for constraining the dimension of the parameter space. Given the view that the random walk is relatively accurate for forecasting macroeconomic time series (in levels), the Minnesota prior is based on shrinking the VAR parameters towards univariate random-walk processes. Moreover, VAR models may be considered as linear approximations of DSGE models. For instance, using the idea that VARs can be used to summarise the statistical properties of both observed time series data and data simulated from a DSGE model, Smith (993) showed how they can serve as a device from which the structural parameters could be estimated and for conducting (indirect) inference; see also Gourieroux, Monfort, and Renault (993). Furthermore, the statespace representation in ()-() can, under certain conditions, be rewritten as an infinite order VAR model; see Fernández-Villaverde, Rubio-Ramírez, Sargent, and Watson (7). If these conditions are not met, then the state-space representation of the DSGE model may have a VARMA representation, where the moving average term is not invertible. An early attempt of combining DSGE models with Bayesian VARs is Ingram and Whiteman (99), who proposed a way of deriving priors for VARs from the economic model; see also DeJong, Ingram, and Whiteman (). This approach was further developed by Del Negro and Schorfheide () into the so-called DSGE-VAR, where the DSGE model is used to determine the moments of the prior distribution of the VAR parameters using an inverted Wishart-normal form. The authors find that this model can compete in forecasting exercises with BVARs based on the Minnesota prior. Similar to the ideas in Smith (993), they demonstrate how posterior inference about the DSGE model parameters can be conducted via the VAR by integrating out the dependence of the VAR parameters from the joint posterior and thereby obtaining a marginal likelihood function for the parameters of the DSGE model; see also Del Negro and Schorfheide (). Moreover, they showed how the DSGE model can be utilised for providing identifying restrictions for the DSGE-VAR, thereby allowing for comparisons of, e.g., impulse responses between the DSGE model and the DSGE-VAR. The DSGE-VAR approach was further enriched by Del Negro, Schorfheide, Smets, and Wouters (7) into a framework for assessing the time series fit of a DSGE model. In this study, we shall consider two classes of BVARs, one that is intended for systems with a smaller dimension and one that has been proposed for large data sets; cf. Bańbura, Giannone, and Reichlin (). The usefulness of BVARs of the Minnesota type for forecasting purposes has long been recognised, as documented early on by Litterman (9), and such models are therefore natural benchmarks in forecast evaluations. While a DSGE-VAR is also a relevant Since the number of shocks and measurement errors of the NAWM is greater than the number of observed variables, the model does not satisfy the conditions in Fernández-Villaverde, Rubio-Ramírez, Sargent, and Watson (7).

15 candidate forecast model, we have opted to focus on BVARs with statistically motivated priors. In addition to models estimated with Bayesian methods, we shall also consider more traditional forecasting models in our empirical exercise. The small BVAR is based on the parameterisation and prior studied by Villani (7). That is, we consider a VAR model with a prior on the steady-state parameters, and a Minnesota-style prior on the parameters on the lags of the endogenous variables; see also Adolfson, Lindé, and Villani (7). For the p-dimensional covariance stationary vector z t the VAR is given by: z t = Ψd t + k ( ) Π l zt l Ψd t l + εt, t =,...,T. () l= The d-dimensional vector d t is deterministic, and the residuals ε t are assumed to be i.i.d. normal with zero mean and positive definite covariance matrix Ω. The Π l matrix is p p for all lags, while Ψ is p d and measures the expected value of x t conditional on the parameters and other information available at t =. One advantage with the parameterisation in () is, as pointed out by Villani (7), that the steady state (or mean) of the endogenous variables is directly parameterised via Ψ. For the standard parameterisation of a VAR model the parameters on the deterministic variables are written as Φ = (I p k l= Π l)ψ when d t =. This makes it difficult to specify a prior on Φ which gives rise to a reasonable prior distribution on the steady state. Moreover, when z t is a subset of the observed variables used in the estimation of the NAWM, we can directly form a prior on the steady state of z t that is consistent with the steady-state prior for the NAWM as captured by a prior on A. This allows for a more balanced comparison between the models since they can share the same prior mean, or steady state, for the variables that appear in both models. The steady state in the NAWM is calibrated, while the steady-state prior covariance matrix is positive definite for the BVAR. Hence, some imbalance between the models remains for the steady-state parameters. Details on the small BVAR model specification are given in Appendix A. Let p(ψ,π,ω Z T ) denote the posterior density, where Π = [Π Π k ] and Z T = {z,...,z T }. Simulation from this distribution is performed via Gibbs sampling for the three groups of parameters Ψ, Π, and Ω using the full conditional posteriors given by Villani (7, Proposition.). Out-of-sample forecasts for the BVAR are calculated for the sample T +,...,T + H, with the objective of estimating the predictive distribution p(z T+,...,z T+H Z T ). The algorithm used for a BVAR was adapted to a multivariate setting by Villani () from the univariate approach suggested by Thompson and Miller (9). That is, () Draw (Ψ,Π,Ω) from p(ψ,π,ω Z T ); () Draw residuals ε T+,...,ε T+H from N(,Ω) and calculate a path for the endogenous variables z T+,...,z T+H using the VAR in (); (3) Repeat step M times for the same (Ψ,Π,Ω); () Repeat steps -3 M times. 5

16 If the forecast evaluation exercise only requires estimates of, e.g., the mean and the covariance matrix of the predictive distribution, the above algorithms need not be used. For example, if the lag order of the VAR is k = and d t =, the mean of z T+h given Z T and the parameters is E [ z T+h Z T,Ψ,Π,Ω ] = Ψ + Π h( z T Ψ ). (7) The mean of z T+h given Z T can therefore be estimated by the average of the right-hand side of (7) over M draws from the posterior of (Ψ,Π,Ω). To estimate the covariance matrix of the predictive distribution we first note that C [ z T+h Z T,Ψ,Π,Ω ] h = Π i Ω ( Π i). () The covariance matrix of z T+h given Z T can now be estimated by adding the sample average of () over M draws from the posterior of the parameters to the sample covariance matrix of (7) over the same draws. The former term measures the part of the h-steps ahead forecast uncertainty due to the VAR innovations, while the latter term reflects parameter uncertainty. In this paper, the variables in the BVAR with a steady-state prior are the same as were used by Smets and Wouters (3), except they are measured as in the NAWM. That is, we use the following variables: real GDP growth, real private consumption growth, real total investment growth, GDP deflator inflation, employment, nominal wage growth, and the short-term nominal interest rate. Hence, two of the variables are given in levels (employment and the short-term nominal interest rate), while the remaining appear in first differences. Bańbura, Giannone, and Reichlin () advocate the use of high-dimensional BVARs for macroeconomic forecasting purposes. Building on the well-known Minnesota prior and its developments (Doan, Litterman, and Sims, 9; Litterman, 9), the authors suggest that as the dimension of the model increases, the overall shrinkage should be stronger; i.e., that the prior should be tighter. Building on this idea, the authors find that the forecasting performance of a small VAR model can be much improved upon by considering a high-dimensional VAR model (3 macroeconomic indicators). Moreover, their results suggest that forecasting performance is already substantially improved when the VAR model has (carefully) selected macroeconomic variables. We will therefore include two large Bayesian VARs that cover the same variables as the NAWM in the study. That is, we let d t = and z t = y t so that p = n in (). Moreover, we reparameterise the deterministic part such that we can use the constant term (Φ) instead of the steady-state term (Ψ). The VAR may therefore be expressed as: k y t = Φ + Π l y t l + ε t, t =,...,T. (9) l= The prior distribution is based on the extension of the usual Minnesota prior to a normal/inverted Wishart, as in Kadiyala and Karlsson (997) and Robertson and Tallman (999), i=

17 and this prior is implemented via dummy observations (see, e.g., Lubik and Schorfheide, ). Additional dummy observations are added through a prior on the sum of the Π l matrices, thereby yielding non-zero prior correlations between the autoregressive parameters (see Sims and Zha, 99). Details concerning the implementation of the dummy observations prior are given in Bańbura, Giannone, and Reichlin (); see also Appendix B below. Two large BVAR models for y t will be studied in the forecast exercises below. These models primarily differ in how the prior mean of the autoregressive parameters is treated. In both models, the prior mean of Π l for all l as well as for the off-diagonal elements of Π are zero. For the diagonal elements of Π, the prior mean is zero in one of the large BVAR models, henceforth the white-noise prior. The second large BVAR sets the prior mean of these diagonal elements equal to unity if the variable is measured in levels, and zero if in first differences. Below we shall refer to this as a mixed prior. Apart from these differences in the treatment of the mean, the priors of the two large BVARs differ only in terms of the numeric value given to the overall tightness hyperparameter; cf. Appendix B. Posterior sampling is straightforward for the large BVAR models. Specifically, the marginal posterior of Ω is inverted Wishart, while the posterior distribution of (Φ,Π) conditional on Ω is normal; see Appendix B for details. To sample from the joint posterior we may therefore use direct sampling; see, e.g., Geweke (5, Chapter.). Since we will compare the forecasting performance of the NAWM with a small BVAR, we shall also estimate a VAR model for the same choice of variables in z t with maximum likelihood, keeping the lag length fixed at the same value as for the BVAR (k = ). Moreover, we shall check how well the DSGE model fares when comparing it to the naïve random walk and mean benchmarks. The mean is here estimated by the within-sample mean of the variables to be forecast. Similarly, we shall consider a random walk in the variables that are forecasted. Below we shall study the forecasting performance for both quarterly and annual changes of (a subset of) the variables that appear in first differences in the NAWM. Hence, the NAWM and the various VAR models do not change with these changes in the forecasted variables (although their forecasts are affected by it), the mean and the random-walk models do change. Accordingly, no matter which criterium is used for evaluating the forecasting performance across annual and quarterly changes, the ranking of the mean and random-walk models relative to the other models is likely to change. 5. Evaluating Forecast Accuracy The forecast performance of the NAWM along with the reduced-form models will be assessed in this section using a rolling procedure where the parameters are estimated up to period T If a variable x t appears in first differences in the NAWM, x t = x t x t, the random-walk model for quarterly changes is simply x t = x t + ǫ t, while the random-walk model for annual changes is x t = x t + ǫ t. The latter model can be rewritten as x t = x t + ǫ t. Similarly, the mean model for quarterly changes is x t = µ q + ǫ t, while the mean model for annual changes is x t = µ a + ǫ t. The latter model can equivalently be expressed as x t = µ a x t x t x t 3 + ǫ t. 7